For the cases (a) and (b), the existence results for the SDRI model in \(\mathcal {C}_{\rm subgraph}\)  and \(\mathcal {C}_{\rm starshaped}\)  can be found for example in [1]}. For (c), the same statements of Theorems REF  and REF  hold with \(\mathcal {X}_\mathcal {G}:=\mathcal {C}_{\rm capillarity}\)  if \(\mathcal {G}=\mathcal {F}\)  or \(\mathcal {X}_\mathcal {G}:=\widetilde{\mathcal {C}}_{\rm capillarity}\)  if \(\mathcal {G}=\widetilde{\mathcal {F}}\)  (note that \(S_u\)  and \(\Gamma \)  are empty in this case).
For (d)-(f), we postpone the analysis to future investigations since some modifications in the proofs is needed to include boundary Dirichlet conditions or fidelity terms of type
\(\kappa \int _{\Omega \setminus K} |u-g|^p dx\) 
